Speed of Light Measurement

A lab manual based on this simulation is available here. This simulation illustrates the theoretical ideas behind the speed of light experiment.

Instructions:
Drag the detector to where the photos are exiting in the end to measure the speed of light (see calculations below).
Set the angular velocity of the rotating mirror to zero to see how the photons travel. You could drag on the rotating mirror to change its angle directly.
As long as the rotating mirror is at rest, the photons always exit at zero angle.
Now turn on the rotating mirror and adjust the "speed of light" (something you cannot do in real life!) and see how it affects the angle of they exit the instrument.
The higher the speed of light, the smaller is the angle of exit.

Explanation:
When the angular velocity of the rotating mirror is zero, a photons travels the same way back, returning to the detector at zero angle of exit.
When the mirror is rotating, a photon who travels to the circular mirror and comes back to the rotating mirror will find the mirror at a different orientation from before due to the finite amount of time it takes to run to the edge and back to the middle. As a result, the photo no longer can return the same way and will exit at a non-zero angle. The lower the speed of light, the more time it allows the mirror to rotate, and the larger is the final angle. Therefore by measuring the angle one can deduce the speed of light.

Remarks:
In the actual experiment, the circular mirror will need to have an opening (not shown in the simulation for simplicity) on the right for the photon to escape.
The "speed of light" in the simulation is very low compare to the actual speed of light, so in practice the rotating mirror will need to rotate at an extremely high rate to achieve just a tiny deflection. Usually the angle of exit is so close to zero that one needs to use a microscope to observe it!

The Relativity of Simultaneity

A lab manual based on this simulation is available here.

• Two events occurred simultaneously from the perspective of the top observer.
• See if they are also simultaneous for the bottom observer based on the light signals from the two events.
• Note that the simulation is in the perspective of the top observer. Therefore due to the length contraction the bottom ship would actually be longer than the ship at the top if they are both at rest.

Relativity: Time Dilation

Adjust the velocity and see how time flows at different rate in the two light clocks.
Note that the photons always travel at the speed of light \(c\).

Length Contraction

Change the velocity to observe length contraction.
You can also drag the two objects together to compare their sizes.

The Necessity of Length Contraction

The view at the top shows two photons bouncing inside an L-shaped ship. The two photons always gets back to the corner at the same time.
The view at the bottom is the same ship viewed from a moving frame. If length contraction is disabled (so the horizontal arm does not contract), the two photons will be coming back out of sync, leading to contradiction.
Therefore length contraction is necessary to perserve the local observation (made at the corner) that the two photons always arrive back at the same time.
Although it may appear to you the photons are moving at different speed, in fact all photons are always traveling at the same speed. The illusion arises because the photons are moving on top of a moving object, thus making some appear to travel faster than others.

Relativity: Ladder and Garage Paradox

A lab manual based on this simulation is available here.

Instructions:
A ladder is too long to fit inside the garage. Can you speed up the ladder to shorten it (length contraction) so it can fit inside the garage at least momentarily?
It appears the plan works... but now click on "Switch Observer" to see how the situation appears from the ladder's perspective. It looks like the plan not only failed, but in fact made matter worse by shortening the garage even more!
So does the ladder fit inside the garage or not when it is moving at high speed?

Explanation:
The key is the relativity of simultaneity, i.e. the idea that "at the same time" is a relative concept.
From the garage's perspective, at some point the doors were both closed "at the same time", trapping the ladder inside. However, what happened at the same time according to the garage did not happen at the same time according to the ladder. In other words, from the ladder's viewpoint the doors were never both closed at the same time.

Even more detailed explanation (if you are interested!):
In relativity the temporal ordering (what happened first) of events is relative. Define two events, LC for the closing of the left door, RO for the opening of the right door. From the garage's perspective, LC happened before RO, so the ladder was momentarily trapped. From the ladder's perspective, RO happened before LC, so by the time the left door closes, the right door is already open, therefore "untrapping" the ladder. See if you can observe the ordering of these events in the simulation.

Addition of Velocity

• Notation: speed of the elf w.r.t. the ground $=v_e$, speed of the arrow w.r.t. the elf $=v_a$.
• The x-axis is in lightseconds. 1 lightsecond is the distance light travels in one second.
• Elf frame: From the elf's point of view, he is stationary. It is the ground and the trees that are moving toward him in the $-x$ direction. He shoots an arrow at the speed $v_a$ from his perspective.
• Lab frame: From an ground observer's (i.e. lab's) point of view, the elf is moving forward toward the trees when he shoots an arrow. The velocity of the arrow w.r.t. the ground is the relativistic sum of the arrow's launch speed and the speed of the elf $\frac{v_e+v_a}{1+ \frac{v_e v_a}{c^2}}$. The relativistic sum is always less than $c$.
• Newtonian frame (wrong!): If Newtonian mechanics is right but relativity is incorrect, then this is what one would see from the ground. The arrow's speed w.r.t. the ground would simply be the sum $v_e+v_a$. This sum could exceed $c$.
• Observe when the Newtonian frame sometimes causes the arrow speed to exceed $c$.
• When the arrow is launched at $c$, see if arrow in the lab frame travels faster than the one in the elf frame. Reduce the launch speed and see if the answer is the same.
• Set the elf's velocity to negative and see which arrow is faster at different launch speed.
• Length contraction and time dilation not drawn in the simulation. Otherwise the trees will appear narrower and closer together from the elf's perspective, and the elf will appear to move in slow motion from the ground's point of view.

Artwork from Spriters Resource, uploaded by sutinoer.

Addition of Velocity with Tanh Function

• This is a quick way to compute the final relativistic sum of the velocities from the last simulation of elf shooting an arrow.
• $v_{final} = \frac{v_e+v_a}{1+ \frac{v_e v_a}{c^2}}$ is equivalent to $\alpha_{final} = \alpha_{elf} + \alpha_{arrow}$ (much simpler!).
• $\beta = \tanh \alpha$ by definition.
• Mathematically, it comes from $\tanh(\alpha_1 +\alpha_2) = \frac{\tanh\alpha_1 + \tanh\alpha_2}{1+\tanh\alpha_1 \tanh\alpha_2}$.
• From a (four dimensional) physics perspective, moving at $\beta$ is essentially the same as rotating the $x$ and $t$ axis by $\alpha$ using $\cosh$ and $\sinh$ functions, just like you would with $\cos$ and $\sin$ when you rotate purely spatial axes.
• Combining $\beta_{arrow/elf}$ and $\beta_{elf/lab}$ is the same as rotating first by $\alpha_{elf/lab}$ and then by $\alpha_{arrow/elf}$.